{
  "id": 528,
  "curve_key": "17953180080518541387621735945923695313106944253681:-68717043565315559285272328066570587842237907351167375288069123293125566921",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-374024585010802945575452832206743652356394671952",
    "79533615237633749172768991731343692332956194038170291602659759576822579"
  ],
  "rank_lower_bound": 16,
  "torsion": [
    2
  ],
  "naive_height": 340.23555517865105,
  "faltings_height": 26.32508441891162,
  "conductor": "799959445620234844233764836050816395509063143013902825458669780365231524648255194970",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "19",
    "41",
    "43",
    "71",
    "103",
    "223",
    "311",
    "359",
    "421",
    "647",
    "821",
    "823",
    "1297",
    "1733",
    "2063",
    "55381",
    "122839",
    "169243",
    "292483",
    "655261",
    "1073507",
    "214659463"
  ],
  "discriminant": "616074787578024409641514665956186314751259819860265274832487459076749882186490124563532971697912341040143120970541122256182623175593574400000000",
  "regulator": "395351602368354651682517.3118480999025217616033483481053452939660063292",
  "points": [
    [
      "354633988358131449619271103/1369",
      "264555143232733788892223630728062411643/50653"
    ],
    [
      "-551689953288724258781091",
      "343461600955400443753544915877486673"
    ],
    [
      "-347580083058768283628663",
      "409322973226376697539940027189395581"
    ],
    [
      "-964684991070760344943036083/5041",
      "135865613573767466872592784898134944709191/357911"
    ],
    [
      "-4136817643797561959143236914915421385624882563/28062063475383244912081",
      "1704466915452137099649935896611656739712969848158704847137084715703799/4700882586351224550655164817781479"
    ],
    [
      "928147996606156310442543350353/3594816",
      "90310967530710161449720544718587470733882603/6815771136"
    ],
    [
      "15357392991007407287383761408564192995877/2171556644801296",
      "1896249262446992531877223732441560511629011421540968146074739/101194461471701180753344"
    ],
    [
      "-6606583589703211818592597304878705060163/17278379269668001",
      "927150979937301251341921630692853702544876555227401530471169/2271195422176487874779249"
    ],
    [
      "1798279841227521606650281597885693/1237843489",
      "70253468815074350414403343163989272573781425480447/43551047473487"
    ],
    [
      "796128099908586356092810785786609023853/9101003803369",
      "22462797459364817091966521952980228160941106256075667857643/27455789960954155403"
    ],
    [
      "-175039495971445347277824860547526252209584907/251263072344072507889",
      "178625014494978145408325669331399421974141828764386465141038326497/3982841271425216581390639102537"
    ],
    [
      "30655127685485349997430334701713441890028545933/133405510852616526124969",
      "3685312005286623109451637495102743802980230307747596660979939465388493/48725988161136409409916464128690453"
    ],
    [
      "1694407830921012978610074057634085365299813/1743081865176678049",
      "1833124509253927231864617883391051450301335229882053736696895433/2301318225126469958022900593"
    ],
    [
      "722489351463251009710355911883948746436809/1410576668205770025",
      "250350609677808075823119313986164823749794573165996579486425113/1675309472617507667010775125"
    ],
    [
      "7275896368033934553232595249768271745680013/9136676641798670449",
      "14787214571985436347392984073277072205628806728248748473413250983/27617374074391070147292119543"
    ],
    [
      "13376404297078074209933352188456814702091186745709/119338303881000319999097041",
      "257515369194219109514743759974126712671870906971683127783612144618886807703/1303676363585851110273652546032357812839"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 16, unconditional: rational 2-torsion point, 16 independent points (Neron-Tate height matrix of rank 16; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 16 (PARI/GP 2.15.4 ellrank, effort 2). Fiber t=-157/648 on branch u=59/37 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).",
  "created_at": "2026-09-02 13:43:39",
  "updated_at": "2026-09-02 13:43:39"
}